From: Nico.Temme@cwi.nl (Nico Temme)
Subject: Fourier on C^infinity
Date: Mon, 9 Apr 2001 19:56:45 GMT
Newsgroups: sci.math.research
Summary: Estimating exponential Fourier integrals with saddle-point methods

In Article 2159 kovarik@mcmail.cis.mcmaster.ca (Zdislav V. Kovarik)
wrote:

> The function
> 
> phi(x) = exp(1/(cos(x) - 1),
> 
> defined as 0 at multiples of 2*pi, is C-infinity but not
> analytic. Its Fourier cosine coefficients a(n) 
> (over [-pi, pi]) will go to 0 faster than any negative 
> power of n, but slower than exponentially. (And they 
> don't seem to be elementary.)
> 
> I would be grateful for a reference about the rate of
> convergence to 0 of these coefficients.
> 
> Cheers, ZVK(Slavek).


A bracket is missing; I assume you mean

phi(x) = exp(1/(cos(x) - 1)).


Integrals of the form (I take the interval [0,2\pi], 
giving the same as [-\pi,\pi]):

a(n) = \int_0^{2\pi} \phi(x) \exp(i n x} dx

can be estimated, for large positive n, by using the saddle
point method, although \phi(x) does not depend on large n.
It has a very dominant behavior near the end points of the
interval. 

The saddle points cannot be calculated exactly, but an 
approximation of one of the relevant saddles x_0, 
that is, a solution of

d/dx(1/(cos(x) - 1) + i n = 0,

is 

x_0 \sim (4/n)^{1/3} \exp{\pi i/6}

At that point the integrand is approximately

\phi(x_0) \exp(i n x_0} \sim 2^{-1/3) 3 n^{2/3} \exp(2\pi i/3} .

There is another saddle point near 2\pi, that is at 
x1 \sim 2\pi - conjugate{x_0}
with a similar value of the integrand.

This gives a first idea about the value of a(n).
The dominant part is \exp(-2^{-4/3) 3 n^{2/3}),
which is exponentially small if n is large.

If one wants more details further analysis is needed.

Integrate with x from 0 to +i\infty through x_0 and back from
+i\infty to 2\pi through x_1. A saddle point contour can be used
(or a local analysis near x_0 and x_1) that will give a full 
asymptotic exapnsion. If n is negative some symmetry
in the Fourier coefficient can be used (or by doing 
the same again in the lower complex plane). 


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